A143491 → basis A008277 → θ+1

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Triangle begins:
  0:         12,          16,           3;
  1:         60,         120,          45,           4;
  2:        360,         960,         540,          96,           5;
  3:       2520,        8400,        6300,        1680,         175,          6;
  4:      20160,       80640,       75600,       26880,        4200,        288,         7;
  5:     181440,      846720,      952560,      423360,       88200,       9072,       441,        8;
  6:    1814400,     9676800,    12700800,     6773760,     1764000,     241920,     17640,      640,       9;
  7:   19958400,   119750400,   179625600,   111767040,    34927200,    5987520,    582120,    31680,     891,    10;
  8:  239500800,  1596672000,  2694384000,  1916006400,   698544000,  143700480,  17463600,  1267200,   53460,  1200,   11;
  9: 3113510400, 22832409600, 42810768000, 34248614400, 14270256000, 3424861440, 499458960, 45302400, 2548260, 85800, 1573, 12;

The row polynomials satisfy (1+(n+2)*x)*P(n,x) = (n+4+(n^2+6*n+10)*x+(n+3)*x^2)*P(n-1,x) + (x+(n+3)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 12 + 16*x + 3*x^2.

P[0, x_] := 12 + 16*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n + 4 + (n^2 + 6*n + 10)*x + (n + 3)*x^2)*P[n - 1, x] + (x + (n + 3)*x^2)*D[P[n - 1, x], x])/(1 + (n + 2)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 12 + 16*x + 3*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((n + 4 + (n^2 + 6*n + 10)*x + (n + 3)*x^2)*v[i - 1] + (x + (n + 3)*x^2)*deriv(v[i - 1], x))/(1 + (n + 2)*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Properties

• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
  by Laguerre’s theorem on multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on multiplier sequences
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

  • from basis_map_A008277 of Unsigned 2-Stirling numbers of the first kind.?
    • Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›

      • from Unsigned 2-Stirling numbers of the first kind.?
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
        • Parent-row provers of polytool polytool ? step › direct check
        • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    • Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›

      • from Unsigned 2-Stirling numbers of the first kind.?
        • ↑ Liu–Wang theorem (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

• A136124 → basis A008277 → θ+1

References

1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
3. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
4. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.